Combinatorics of injective words for Temperley-Lieb algebras

Rachael Boyd; Richard Hepworth

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arXiv · 2006.04261

Combinatorics of injective words for Temperley-Lieb algebras

Rachael Boyd·Richard Hepworth

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Abstract

This paper studies combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of certain standard Young tableaux. This trio of results - inspired by results of Reiner and Webb for the complex of injective words - can be viewed as an interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue of the number of derangements of n letters. This interpretation has precursors in the literature, but here emerges naturally from considerations in homological stability. Our final result shows a surprising connection between the boundary maps of our complex and the Jacobsthal numbers.

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Rachael Boyd, Richard Hepworth. 2020-06-07. Combinatorics of injective words for Temperley-Lieb algebras. https://arxiv.org/abs/2006.04261

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